On the representability of the biuniform matroid
Date of Issue2013
School of Physical and Mathematical Sciences
Every biuniform matroid is representable over all suﬃciently large ﬁelds. But it is not known exactly over which ﬁnite ﬁelds they are representable, and the existence of eﬃcient methods to ﬁnd a representation for every given biuniform matroid has not been proved. The interest of these problems is due to their implications to secret sharing. The existence of eﬃcient methods to ﬁnd representations for all biuniform matroids is proved here for the ﬁrst time. The previously known eﬃcient constructions apply only to a particular class of biuniform matroids, while the known general constructions were not proved to be eﬃcient. In addition, our constructions provide in many cases representations over smaller ﬁnite ﬁelds.
SIAM journal on discrete mathematics
© 2013 Society for Industrial and Applied Mathematics (SIAM). This paper was published in SIAM Journal on Discrete Mathematics and is made available as an electronic reprint (preprint) with permission of SIAM. The paper can be found at the following official DOI: [http://dx.doi.org/10.1137/120886960]. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law.